I saw something that I need a little bit of help with understanding the use of the genitive. I understand that there are two basic ways of indicating possession:
1. Моята книга (my book); Моя книга (a book of mine)
2. Книгата ми (my book)
…but I keep seeing the placement, in the case of No.2, in different places.

For example:
"Майка ти целунала ми баща!" and "Кажи й, че съм ти сестра" and even "Той не ми разбра книгата" (First and last from Facebook and the one in middle is from a song).

Can you understand good Bulgarian?
I will explain it at first in English. Кажи й, че съм ти сестра. is not genitiv, however dativ. It is a shortened form from: Кажи й на нея, че.....But the first one is much better and mostly used in speech, and not only. You put the pronoun after a verb. Examples:Книгата му (his book), разказах му (I told him), докладвам му (I report him). In Bulgarian there are no rules where to put a specific word. This can be seen here: Той не ми разбра книгата. As I already said, it could be, as well: Той не разбра книгата ми.

When it is the DATIVE case, you place the pronoun after the VERB (kazah mu). When it is Genitiv CAse you put the pronoun after the noun (drehata i). Thats why it is also possible to say: Той не разбра книгата ми.

This is a bit complicated, so I'll try to break it down. When possession is expressed with pronouns, one has two options in Bulgarian - Dativus possessivus and regular Genitive. The constructions corresponding to the Dative and Genitive case are merged in Bulgarian (this a feature of the Balkan Sprachbund). We use the same preposition "на" to introduce the indirect object of a verb and to introduce the possessor of something, also, the short dative and short genitive pronouns are the same. Kinship terms and copula constructions are a special case.

Position of the short genitive pronouns in the noun phrase

Unless we are dealing with kinship terms, the short genitive pronouns are used only with definite noun phrases, that is, the definite article has to be present in the noun phrase. The definite article is put only on the first constituent of the noun phrase that can take it, the pronoun is put immediately afterwards. The parts of speech than can take the definite article are: nouns, adjectives, ordinal and cardinal numbers, the following participles: present active, present passive, past active aorist and past passive. Additionally, some adverbs of quantity can also take the definite article (многото, малкото, повечето), the same is true for some pronouns of quantity, namely the indefinite pronouns "няколкото", "неколцината", the negative pronoun "николкото", and the summative pronouns "всичкият" (всичката, всичкото, всичките).

The constituents of the noun phrase are ordered thus: adverb of quantity/demonstrative/indefinite/negative/ or summative pronoun - number - adverb modifying an adjective - adjective - noun.

Examples (in all examples the pronoun is after the constituent that takes the definite article):

Unlike in English, in Bulgarian we can have both a demonstrative pronoun and a possessive pronoun in the same noun phrase. In English one can't say "this my book". If the definiteness of the noun phrase is expressed not with a definite article, but with a demonstrative pronoun, the short genitive pronoun goes after it. In this case the short and long possessive pronouns can be used interchangeably:

When you use the regular Genitive (G), the position of the short possessive pronouns is determined by the above rules. If you use dativus possessivus (DP), the position of the short dative pronoun is determined by the rules of clitic placement, which are also a bit complex and we may need to open a separate thread for them, because we need to deal with a whole bunch of other things, such as short accusative pronouns, present forms of the verb "to be", question words, etc. When you use dativus possessivus, the noun phrase has to be used with the definite article.

Generally speaking the constructions are interchangeable, one word order may be preferred over another, depending on which part of the sentence you want to put the emphasis on. So you can say:

Be careful when the noun phrase is the complement of a preposition and the verb can also have an indirect object. In such cases it is better not to use DP because the result may be ambiguous. Let's take the verb "пиша" (to write) for example. It can have both a direct and an indirect object - to write a lettter (direct object) to John (indirect object). So the sentence "Тя ми пише с молива" (the noun phrase "молива" is the complement of the preposition "с") could mean "She's writing with my pencil", but it can also mean "She's writing to me with the pencil". To be safe use regular genitive - "Тя пише с молива ми".

If the verb is intransitive, which means that it cannot have a direct or indirect object, like "to sit", or if it's used intransitively, like "to drink from", then we do not have a problem and we can use DP:

You are sitting on my seat:

Седиш ми на мястото (DP, this variant is more natural because the noun phrase is short)Седиш на мястото ми (G)

Remember that when we use Dativus Possessivus, the noun phrase has to be definite, so we might have problems with those kinship terms that don't take the definite article. In such cases we have to use the regular genitive:

You can't say "Болен ми е брат" (My brother is ill) because "брат" is indefinite, neither can you say "Болен ми е братът" because "брат" doesn't take the definite article. You have to say "Болен е брат ми" or "Брат ми е болен". If we alter the noun phrase a bit, so that it takes the definite article, we can use DP - Болен ми е по-големият брат (My elder brother is ill).

We don't have that problem with "дете" (child) because it does take the definite article and we can use Dativus Possessivus:

My child is ill:Болно ми е детето.

as well as regular genitive:

Детето ми е болно.Болно е детето ми.

Having said that, the example that you give "Майка ти целунала ми баща", which attempts to use DP, is not grammatically correct for two reasons. First, the position of "ми" after the verb is justifiable only if there is a pause before the verb (Майка ти, целунала ми баща, but this still isn't correct), second, if we use DP, we have to use the definite article, but "баща" does not take it (you can't say "Майка ти ми целунала бащата"). So the only way of saying that sentence is with regular genitive - "Майка ти целунала баща ми" (You mother reportedly kissed my father)

I'm having trouble explaining the other example that you've given "Кажи й, че съм ти сестра", because that's definitely a dative construction, but the noun phrase is not definite, so I don't think it's Dativus Possessivus. I'll try to explain what the difference between "Аз съм сестра ти" and "Аз съм ти сестра" is, they both mean "I'm your sister".

"Аз съм сестра ти" implies some kind of definiteness. I'm not just any sister of yours, I'm that particular sister of yours, the one you lost and whom you've been looking for.

"Аз съм ти сестра" just explains what our relationship is, a literal translation would be "I am a sister to you". It looks very much like the Latin sentence "Homo homini lupus est" - "a man is a wolf to a man", which also uses a dative construction.

In conclusion, in Bulgarian we can express possession also with the dative case. To know where to put the short dative pronouns, you need to learn about the clitics placement, which include other pronouns, some verb forms and some particles.

Exactly, but it sounds too old fashioned and almost no one would say anything like that. I am from West Bulgaria and these usage of the DATIVE POSSESSIVUS is not used at all and for me it is even odd and weird. Better would be to use the GENITIVE: че съм твоя сестра (instead:че съм на теб сестра.).The best way to express yourself is to use the shortened forms of the possessive pronouns (GENITIVE). SO: че съм тисестра or че съм сестра ти. In general DATIVE POSSESSIVUS is not used and in the bulgarian grammar books it is only mentioned.
PC: Виждам, че си българин по това че така си написал и по това че ми отговори на една тема, която пуснах преди време. За което искрени благодарности. Ти къде живееш всъщност? Ако в България - къде по - точно?
In addition, I extra asked my grandmother, who comes from Pleven (North Bulgaria), and she told me that she also prefers the second sentence. She said as well , that she would also use the archaic DATIVE CASE (I reckon its dative; not sure): че съм сестра твоему. But it is too dialectic and old.

I think we shouldn't confuse the readers of this thread. DATIVUS POSSESSIVUS is used quite often in Bulgarian. The most natural way of saying "I'm washing my hands" in Bulgarian is by using DP: "Мия си ръцете". DP is used in many other languages:

твоему is masculine, сестра is feminine, they have to agree. Besides in the sentence "че съм твоя сестра" (that I'm your sister), "твоя сестра" is the subject complement, not the indirect object, so it should be in the nominative case, not the dative.

Exactly, but it sounds too old fashioned and almost no one would say anything like that. I am from West Bulgaria and these usage of the DATIVE POSSESSIVUS is not used at all and for me it is even odd and weird. Better would be to use the GENITIVE: че съм твоя сестра (instead:че съм на теб сестра.).The best way to express yourself is to use the shortened forms of the possessive pronouns (GENITIVE). SO: че съм тисестра or че съм сестра ти. In general DATIVE POSSESSIVUS is not used and in the bulgarian grammar books it is only mentioned.
PC: Виждам, че си българин по това че така си написал и по това че ми отговори на една тема, която пуснах преди време. За което искрени благодарности. Ти къде живееш всъщност? Ако в България - къде по - точно?
In addition, I extra asked my grandmother, who comes from Pleven (North Bulgaria), and she told me that she also prefers the second sentence. She said as well , that she would also use the archaic DATIVE CASE (I reckon its dative; not sure): че съм сестра твоему. But it is too dialectic and old.